A New Algorithm for Finding Minimal Cycle-Breaking Sets of Turns in a Graph

Abstract

We consider the problem of constructing a minimal cycle-breaking set of turns for a given undirected graph. This problem is important for deadlock-free wormhole routing in computer and communication networks, such as Networks of Workstations. The proposed Cycle Breaking algorithm, or CB algorithm, guarantees that the constructed set of prohibited turns is minimal and that the fraction of the prohibited turns does not exceed 1/3 for any graph. The computational complexity of the proposed algorithm is O(N²∆), where N is the number of vertices, and ∆ is the maximum node degree. The memory complexity of the algorithm is O(N∆). We provide lower bounds on the minimum size of cycle-breaking sets for connected graphs. Further, we construct minimal cycle-breaking sets and establish bounds on the minimum fraction of prohibited turns for two important classes of graphs, namely, t-partite graphs and graphs with small degrees. The upper bounds are tight and demonstrate the optimality of the CB algorithm for certain classes of graphs. Results of computer simulations illustrate the superiority of the proposed CB algorithm as compared to the well-known and the widely used Up/Down technique.